Analysis of many engineering problems involve obtaining the solution of complex governing differential equations in a domain of interest. For example, thermal analysis involves the solution of a three dimensional heat conduction equation, fluid flow analysis involves the simultaneous solution of conservation of mass and Navier Stokes equations, and structural and vibration analysis involve the solution of equations based on conservation of momentum and energy. Due to the degree and order of these governing equations, analytical solutions are feasible only for simple geometries and cases involving one and two dimensions. For more generic problems involving complex three dimensional geometries and heterogeneous materials, analytical solutions to the governing differential equations are impractical. Consequently, various pre-existing numerical techniques have been developed for the purpose of solving differential equations. In this regard, pre-existing finite difference methods, finite element methods and boundary element methods are widely used to solve many complex engineering problems. However, while these various pre-existing techniques have been generally adequate for their intended purposes, they have not been satisfactory in all respects
In more detail, finite difference methods use pointwise approximations with differential formulation. In other words, approximate solutions are obtained at several points in the region or domain of interest, and governing differential equations are transformed into algebraic equations using finite difference approximation. However, due to the use of pointwise approximation, the solutions have acceptable accuracy only at the specific points selected within the region of interest. Moreover, finite difference methods cannot accurately represent complex geometries, due to the use of pointwise approximation.
Finite element and boundary element methods use piecewise approximations with integral formulation. In other words, a region of interest is approximated within finite regions called finite elements, and these approximations are then used to transform governing differential equations into algebraic equations with standard integral formulation methods, such as a variational approach, weighted integrals, virtual work, or the like. Although these methods can represent complex geometries and can yield accurate results, integral formulations are very complicated, and even computerized computation of a solution takes a long time and consumes a lot of memory.